Fast Method for Excited-State Dynamics in Complex Systems and Its Application to the Photoactivation of a Blue Light Using Flavin Photoreceptor

The excited-state dynamics of molecules embedded in complex (bio)matrices is still a challenging goal for quantum chemical models. Hybrid QM/MM models have proven to be an effective strategy, but an optimal combination of accuracy and computational cost still has to be found. Here, we present a method which combines the accuracy of a polarizable embedding QM/MM approach with the computational efficiency of an excited-state self-consistent field method. The newly implemented method is applied to the photoactivation of the blue-light-using flavin (BLUF) domain of the AppA protein. We show that the proton-coupled electron transfer (PCET) process suggested for other BLUF proteins is still valid also for AppA.

In iMOM 1 , reference orbitals are that of the excited guess determinant and new occupied orbitals are selected following a similarity criterion. Let us define the overlap O between current MOs and reference MOs as (S1) The sum over the row index of the overlap matrix O produces p j , which indicates the projection of the j-th new orbital onto the space of old occupied orbitals: In eq. S2, i and j are occupied MOs indices and µ and ν refer to atomic orbital basis functions.
New occupied orbitals are selected as the orbitals that have the highest values of p j , which means orbitals that are "similar" to reference orbitals. The advantage of iMOM is that the information contained in the guess, namely the reference determinant that is orthogonal to the ground state and that targets a specific excitation, is retained through the iterative optimization. Therefore, no gradual collapse of the wavefunction onto the ground state is possible.
STEP was presented by Carter-Fenk et al. 2 as a robust and efficient alternative to iMOM. This is a completely different approach because it employs the Aufbau principle to choose new occupied S1 orbitals. However, the constraint which allows the solution to remain in the excited state is a level shift of the virtual orbital energies 2 . Let us consider the operatorQ, which is a projector onto the This projector can be expressed in the atomic orbital basis set as follows: The level shift is performed by adding the matrix form of eq. S4 to the Fock (or Kohn-Sham) matrix, with a parameter η which establishes the amount of level shifting. The new Fock matrix F is written in eq. S5 and replaces the conventional Fock matrix during the whole iterative procedure.
This expedient can be used to facilitate the convergence to the excited state solution when an excited determinant is given as a guess. Indeed, if the energy of virtual orbitals is increased such that all virtual orbital energies of the excited guess determinant are greater then all the occupied ones, the desired set of occupied orbitals is simply retained with the Aufbau occupation. At the first step, the projectorQ is chosen on the basis of the reference orbitals, which are usually the same as the ground-state orbitals but with a different occupation targeting an excited state.
Note that the level shift parameter η must be carefully chosen, because a high level shift can reduce excessively the occupied-virtual orbital rotations, resulting in a slower convergence.
where is an empirical parameter and it is often set to 0.1 Hartree.

S2 Details on the Grassmann extrapolation for open-shell systems
In this section we briefly present the details of the generalization of Grassmann extrapolation 3 to open-shell systems. This allows us to use this procedure for reducing the computational cost of ∆SCF/MM excited-state simulations. The theory behind the extrapolation can be easily extended to unrestricted SCF by working separately on α and β electrons. The new extrapolation problem can be formulated as follows: there are two sets of pairs (R i , D α i ) and (R i , D β i ) and a new position vector R n . The gaol is to guess density matrices D α R and D β R for that geometry.
In the scheme, R is the current geometry and d R are molecular descriptors (we use the Coulomb matrix). As for the closed-shell case, we find the coefficients c R that provide the best approximation of d R , and we use these coefficients to perform a linear combination of vectors in the tangent space. There are two tangent spaces, identified by the Grassmann Logarithm based on D α 0 and D β 0 , which are α and β density matrices at the first MD step. Γ α R and Γ β R are the results of linear combinations in the tangent spaces. Extrapolated density matrices (D α R and D β R ) are recovered by means of Grassmann Exponential mappings.
Ideally, this extrapolation procedure produces converged density matrices at the current step of simulation. Since we perform some approximations, we can state that D α R and D β R are the best approximation of converged density matrices, and we can use these matrices as a guess for ∆SCF iterations.